A note on "New classes of iterative methods for nonlinear equations'' and "Some iterative methods free from second derivatives for nonlinear equations''

作者：Rafiq Arif^*

来源：Applied Mathematics and Computation, 2007, 193(2): 572-576.

DOI：10.1016/j.amc.2007.04.054

• iterative methods
• two-step method
• nonlinear equations
• convergence order
• numerical examples

In [M. A. Noor, New classes of iterative methods for nonlinear equations, Appl. Math. Comput., in press; M. A. Noor, Some iterative methods free from second derivatives for nonlinear equations, Appl. Math. Comput., in press], Noor introduced a generalized one parameter Halley's method
x(n+1) = x(n) - f(x(n))f'(2)(x(n))/f'(3)(x(n)) - alpha f(x(n))f ''(x(n))
for solving the nonlinear equation f(x) = 0. Noor further showed that for alpha = 1/2 f'(x(n)), the above method reduces to the Halley's method [E. Halley, A new exact and easy method for finding the roots of equations generally and without any previous reduction, Philos. Roy. Soc. London 18 (1964) 136 - 147]. It is interesting to note that for alpha = f'(3)(x(n))/f(x(n))f ''(x(n)), the above method fails. In this note, we point out some major bugs in the results of Noor (in press).

• 出版日期2007-11-1